In
theoretical computer science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
It is difficult to circumsc ...
and
mathematics, computational complexity theory focuses on classifying
computational problem
In theoretical computer science, a computational problem is a problem that may be solved by an algorithm. For example, the problem of factoring
:"Given a positive integer ''n'', find a nontrivial prime factor of ''n''."
is a computational probl ...
s according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
.
A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical
models of computation
In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes ho ...
to study these problems and quantifying their
computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...
, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in
communication complexity), the number of
gates in a circuit (used in
circuit complexity) and the number of processors (used in
parallel computing). One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do. The
P versus N